The wikipedia page http://en.wikipedia.org/wiki/Sphere_packing gives most of the information, including Kepler's conjecture and a reference to Laszlo Toth's exhaustive proof about maximum density packing, so I'll take that for granted.This problem is different, because Kepler's conjecture, and the proof, assume an infinite space, and we have a 96 in. circumference sphere bounding the space. The clipping of ping pong balls becomes an interesting problem to express mathematically, except by some summations with certain criteria and constraints.I assume we are looking for the "maximum" number, to make the problem more interesting, and that we speak of "whole" and "spherical" ping pong balls.The problem with the information wikipedia page, is that it describes three layers, A, B, C, which can give the same packing density, but which may result in different "clipping" of ping pong balls near the edge.For example, two common packing methods, face centered cube, and hexagonal close packing, give the same density, but FCC is composed of layers A, B, C, while HCP is composed only of layers A, and B. But any combination of layers that doesn't repeat give equal packing densities, yet different clipping.Another wikipedia page http://en.wikipedia.org/wiki/Close-packing_of_spheres gives a method to generate x,y,z coordinates of the ping pong balls.Using that, I wrote a small script to iterate over the x, y, z directions, calculating the coordinates, and counting how many ping pong balls fit within the distance encased by the larger sphere (i.e. sqrt(x*x + y*y + z*z) + r < R). I noted that different packing strategies yielded different results.Assuming my calculations are correct, my results are summarized below, using the A, B, C layers described in the wikipedia pages. You need to understand that for the following descriptions to make sense.I calculated a theoretical upper bound (where I let layers repeat) at 4635 ping pong balls.The FCC packing (BC ABC ABC ABC ABC ABC ABC ABC) yielded 4585 ping pong balls.The FCC packing, starting with layers A or C, yielded 4577 ping pong balls.A different FCC packing (hard to describe with ABC, except that the balls were moved diagonally so that a hole was in the center) only yielded 4508 ping pong balls.The HCP packing (AB AB AB AB AB AB AB AB AB AB AB A) yielded 4602 ping pong balls.While HCP packing starting with layer B, yielded 4559 ping pong balls.The best packing I could find (AB AB AB ACB AB AB ABC AB AB AB A) yielded 4631 ping pong balls.
mcguirkthejerk wrote:
Okay letsrun brainiacs, I have a tough question and I want to see who can solve it:
I want to know how many ping pong balls you can fit within a much larger sphere.
The larger sphere has a circumference of ~96in.
A ping pong ball has a circumference of ~4.9634in.
You do the math and as always, show your work.
Anyone who comes up with the correct answer is up for a prize.